Kernel and Image in Linear Transformations
Welcome, students π In this lesson, you will learn two of the most important ideas in linear transformations: the kernel and the image. These ideas tell us what a transformation destroys and what it creates. They are central to understanding how matrices and functions act on vectors in geometry, science, computer graphics, data analysis, and more.
What you will learn
By the end of this lesson, you should be able to:
- explain the meaning of the kernel and image of a linear transformation,
- find the kernel and image using examples and procedures,
- connect these ideas to the larger study of linear transformations,
- describe why kernel and image matter in linear algebra,
- use evidence from vectors and matrices to support your reasoning.
Why kernel and image matter
Imagine a machine that takes in a vector and outputs another vector. This machine is called a linear transformation. If you know what inputs go to zero, and what outputs are possible, you understand a lot about the machineβs behavior. That is exactly what the kernel and image help us measure. π―
For example, in a video game, a transformation might stretch, rotate, or flatten objects. Some directions might disappear completely, while other directions remain visible in the output. The kernel captures the disappeared inputs, and the image captures all possible outputs.
The kernel: inputs that go to zero
The kernel of a linear transformation is the set of all vectors that are sent to the zero vector. If $T$ is a linear transformation, then its kernel is written as
$$\ker(T)=\{\mathbf{v} \mid T(\mathbf{v})=\mathbf{0}\}$$
This means the kernel contains every input vector that becomes invisible after the transformation.
A simple idea
Suppose a transformation $T$ takes vectors in $\mathbb{R}^2$ and projects them onto the $x$-axis. Then any vector of the form $\begin{bmatrix}0\y\end{bmatrix}$ gets mapped to $\begin{bmatrix}0\\0\end{bmatrix}$. Those vectors are in the kernel because they disappear completely.
So, for that projection, the kernel is
$$\ker(T)=\left\{\begin{bmatrix}0\y\end{bmatrix} : y\in\mathbb{R}\right\}$$
This shows that the kernel can contain more than just the zero vector. In fact, every linear transformation always sends the zero vector to the zero vector, so $\mathbf{0}$ is always in the kernel.
Why it matters
The kernel tells us whether a transformation is one-to-one. If the only vector in the kernel is the zero vector, then different inputs cannot collapse to the same output. In that case, the transformation is one-to-one.
If the kernel contains nonzero vectors, then at least two different inputs produce the same output. Why? Because if $T(\mathbf{v})=\mathbf{0}$ for some nonzero $\mathbf{v}$, then for any vector $\mathbf{x}$,
$$T(\mathbf{x}+\mathbf{v})=T(\mathbf{x})+T(\mathbf{v})=T(\mathbf{x})+\mathbf{0}=T(\mathbf{x})$$
So the inputs $\mathbf{x}$ and $\mathbf{x}+\mathbf{v}$ have the same output.
The image: all possible outputs
The image of a linear transformation is the set of all vectors that can be produced by the transformation. If $T$ is a linear transformation, its image is written as
$$\operatorname{αα}(T)=\{T(\mathbf{v}) \mid \mathbf{v}\in V\}$$
The image is also called the range of the transformation.
A simple idea
If a transformation sends vectors from $\mathbb{R}^2$ to $\mathbb{R}^2$ by projecting onto the $x$-axis, then every output has the form
$$\begin{bmatrix}x\\0\end{bmatrix}$$
So the image is the set of all vectors on the $x$-axis:
$$\operatorname{im}(T)=\left\{\begin{bmatrix}x\\0\end{bmatrix} : x\in\mathbb{R}\right\}$$
This means the transformation cannot produce any vector with a nonzero second coordinate.
Why it matters
The image tells us which outputs are actually reachable. Even if the codomain is large, the image may be smaller. This helps answer questions like:
- Which outputs can the transformation make?
- Is the transformation onto?
- How much of the output space is covered?
If the image equals the whole codomain, the transformation is onto or surjective.
Working with matrices
Many linear transformations are represented by matrices. If a matrix $A$ defines a transformation $T(\mathbf{x})=A\mathbf{x}$, then:
- the kernel is all vectors $\mathbf{x}$ such that $A\mathbf{x}=\mathbf{0}$,
- the image is the set of all vectors of the form $A\mathbf{x}$.
For a matrix transformation, the image is also the column space of the matrix. That means the image is the set of all linear combinations of the columns of $A$.
Example 1: Finding the kernel
Consider
$$A=\begin{bmatrix}1 & 2\\2 & 4\end{bmatrix}$$
To find the kernel, solve
$$A\mathbf{x}=\mathbf{0}$$
Let $\mathbf{x}=\begin{bmatrix}x\y\end{bmatrix}$. Then
$$\begin{bmatrix}1 & 2\\2 & 4\end{bmatrix}\begin{bmatrix}x\y\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}$$
This gives the equation
$$x+2y=0$$
The second row is the same equation, so the solutions are
$$x=-2y$$
Let $y=t$. Then
$$\mathbf{x}=\begin{bmatrix}-2t\t\end{bmatrix}=t\begin{bmatrix}-2\\1\end{bmatrix}$$
So the kernel is
$$\ker(T)=\operatorname{span}\left\{\begin{bmatrix}-2\\1\end{bmatrix}\right\}$$
This kernel is not just $\mathbf{0}$, so the transformation is not one-to-one.
Example 2: Finding the image
Using the same matrix,
$$A=\begin{bmatrix}1 & 2\\2 & 4\end{bmatrix}$$
its columns are
$$\mathbf{c}_1=\begin{bmatrix}1\\2\end{bmatrix}, \quad \mathbf{c}_2=\begin{bmatrix}2\\4\end{bmatrix}$$
Notice that
$$\mathbf{c}_2=2\mathbf{c}_1$$
So the two columns point in the same direction. The image is all multiples of $\mathbf{c}_1$:
$$\operatorname{im}(T)=\operatorname{span}\left\{\begin{bmatrix}1\\2\end{bmatrix}\right\}$$
That means every output lies on a line through the origin in $\mathbb{R}^2$.
Rank and nullity connection
The dimension of the image is called the rank of the transformation, and the dimension of the kernel is called the nullity. These quantities are connected by the rank-nullity theorem:
$$\dim(V)=\dim(\ker(T))+\dim(\operatorname{im}(T))$$
This tells us that the size of the input space is split between what is lost in the kernel and what survives in the image.
For a transformation $T: \mathbb{R}^n \to \mathbb{R}^m$ represented by an $m\times n$ matrix, the theorem becomes:
$$n=\operatorname{nullity}(T)+\operatorname{rank}(T)$$
This is one of the most useful ideas in linear algebra because it connects structure, dimension, and solvability.
How kernel and image fit together
The kernel and image describe two different sides of the same transformation.
- The kernel describes what gets sent to zero.
- The image describes what outputs can be reached.
Together, they help answer big questions:
- Is the transformation one-to-one?
- Is the transformation onto?
- How many independent directions survive?
- What information is lost?
For example, if a transformation compresses a plane onto a line, then some directions in the plane vanish. Those vanished directions form the kernel. The outputs that remain are all on that line, which is the image.
A real-world connection
Think about taking a 3D object and projecting its shadow onto a wall. The shadow is the image: it contains every possible point the projection can produce. The directions pointing directly toward the wall disappear in the projection; those are part of the kernel. π
This shows why kernel and image are useful in real life. They help describe compression, data loss, and dimensional reduction in fields like computer graphics, physics, and engineering.
Conclusion
Kernel and image are two core ideas in linear transformations. The kernel tells us which inputs disappear, while the image tells us which outputs are possible. They help us determine whether a transformation is one-to-one or onto, and they reveal the structure hidden inside matrices. students, if you can find the kernel and image of a transformation, you can understand a major part of how linear algebra works.
Study Notes
- The kernel of a linear transformation $T$ is the set of all vectors $\mathbf{v}$ such that $T(\mathbf{v})=\mathbf{0}$.
- The image of $T$ is the set of all vectors that can be written as $T(\mathbf{v})$ for some input vector $\mathbf{v}$.
- The image is also called the range.
- For a matrix transformation $T(\mathbf{x})=A\mathbf{x}$, the kernel is the solution set of $A\mathbf{x}=\mathbf{0}$.
- For a matrix transformation, the image is the column space of $A$.
- If the kernel contains only the zero vector, then the transformation is one-to-one.
- If the image equals the whole codomain, then the transformation is onto.
- The dimensions of the kernel and image are called nullity and rank.
- The rank-nullity theorem states that $\dim(V)=\dim(\ker(T))+\dim(\operatorname{im}(T))$.
- Kernel and image help explain what information a linear transformation loses and what outputs it can produce.
- In geometric terms, the kernel is what gets flattened or erased, and the image is the shape that remains.